1.1. Field of the Invention
This invention relates to improved techniques for determining orbital data of space traveling objects such as earth satellites, and more particularly to improved radio interferometric methods and instrumentation for determining such data.
Orbital data are data representative of the path of a satellite in space and, more specifically, of the position of a satellite at a particular time or as a function of time. Orbital data may represent an orbit in various ways. For example, a satellite's position and velocity vectors may be specified in rectangular coordinates at a certain epoch. Alternatively, the elements of an osculating or a mean ellipse may be given.
Radio interferometric data such as differences of carrier phase observations of satellite signals from a pair of receiving stations constitute a kind of orbital data. However, the present invention concerns the combination of carrier phase data from three or more receivers and the processing of the combined data to determine data more directly representative of the path or position of a satellite in space. Therefore, the term "orbital data" will be used herein to refer to the latter data, and the term "orbit determination" will be used to refer to the process of deriving such orbital data from the phase measurement data.
Although the invention is disclosed with reference to the satellites of the NAVSTAR Global Positioning System, or "GPS", it applies as well to the determination of orbital data of other space traveling, radio transmitting objects, such as the Soviet GLONASS satellites and certain other space craft.
1.2. The Global Positioning System
The GPS is now in the process of being deployed by the U.S. Department of Defense, and will be used mainly for purposes of navigation and position determination. About seven satellites of the GPS now orbit the earth and transmit radio signals by which users can determine their positions on earth.
1.2.1. GPS Satellite Orbits
When complete, the Global Positioning System is expected to include about 21 satellites orbiting the earth in three planes. About seven satellites will be distributed around a geocentric circular orbit in each of these planes; each plane will be inclined to the earth's equator by an angle of about 55 degrees; and the equator crossing points, or nodes, of the orbits will be about equally spaced in longitude, about 120 degrees apart.
The altitudes of the orbits above the surface of the earth are all about 20,000 km, and the common orbital period is about 24 hours as viewed from the rotating earth. Thus, the GPS satellites are not "geostationary", but each appears to a ground-based observer to rise, move through the sky, and set daily. From any given point on the earth's surface, at least four satellites will be in view at any time, 24 hours per day. Because the orbits are so high, a given satellite at a given time may be seen from widely separated points on the earth's surface.
1.2.2. Transmitted Signals
Each GPS satellite transmits microwave L-band radio signals continuously in two frequency bands, centered at 1575.42 MHz and 1227.60 MHz and known as the "L1" and the "L2" bands, respectively. Within each of these GPS bands, the transmitted signal is a broadband, noise-like, pseudorandom signal which contains no discrete spectral components. The signals are therefore said to be carrier-suppressed.
1.2.2.1. Carriers and Modulation
The term "carrier" is used herein in the same sense as is usual in the radio art; that is, a carrier is a periodic wave of essentially constant amplitude, frequency, and phase. Information may be conveyed, or "carried" by varying the amplitude, frequency and/or phase of such a signal. A carrier may be called a "subcarrier" if its frequency is less than the bandwidth of the signal. A signal may include several carriers. For example, a broadcast television signal is said to include a video carrier and an audio carrier.
Although no carriers are present in the GPS signals as transmitted, various carriers may be said to be implicit therein, in that such carriers may be recovered or reconstructed from the GPS signals.
Within each GPS satellite, a stable frequency standard such as an atomic cesium beam device provides a fundamental frequency of 5.115 MegaHertz, called f.sub.0, from which all other critical satellite frequencies are derived by integer multiplication or division. The frequency of the L1band center frequency carrier of GPS signals is 308 times f.sub.0 or 1575.42 MegaHertz, and the frequency of the L2 band center frequency carrier is 240 times f.sub.0 or 1227.60 MegaHertz. The f.sub.0 fundamental frequency is a carrier frequency which may be reconstructed from the GPS signals.
GPS signals are bi-phase or quadriphase modulated. In particular, quadrature components of an L band center frequency carrier are multiplied, in the satellite, with pseudorandom, binary valued waves m(t) and n(t). The m(t) and n(t) waveforms are aperiodic, but periodic carrier waves are implicit in them. Polarity or phase reversals of m(t) and n(t) occur only at times which are integer multiples of fixed time intervals tm and tn known as the chip widths of m(t) and n(t), respectively.
If m(t) reversed Polarity at every multiple of tm, then m(t) would be a periodic square wave with a frequency equal to 1/(2tm). Because the polarity reversals actually occur pseudorandomly, just half the time on average, the 1/(2tm) frequency carrier wave is suppressed, as is the band center frequency carrier.
Similarly, if n(t) reversed polarity at every multiple of tn, then n(t) would be a periodic square wave with a frequency equal to 1/(2tn). Again, because n(t) reverses polarity pseudorandomly, both the 1/(2tn)-frequency carrier and the band center frequency carrier are suppressed.
In each GPS satellite's transmitter, one quadrature component of the 308 f.sub.0 or 1575.42 MHz, L1 band center frequency carrier is modulated with m(t), which has chip width tm equal to 5/f.sub.0 or about 977.5 nanoseconds. The orthogonal component of the L1 band center frequency carrier is modulated with n(t), which has chip width tn equal to 1/(2f.sub.0) or about 97.75 nanoseconds. The 240 f.sub.0 or 1227.60 MHz, L2 band center frequencY carrier is modulated with only n(t). Thus, in the spread spectrum signal transmitted in the L1 band, at least three different carrier waves are implicitly present, with frequencies of f.sub.0 /10 (equal to 0.5115 MHz), f.sub.0 (equal to 5.115 MHz), and 308 f.sub.0 (equal to 1575.42 MHz). In the spread spectrum signal transmitted in the L2 band, at least two different carrier waves are implicitly present, with frequencies of f.sub.0 (5.115 MHz) and 240 f.sub.0 (1227.60 MHz).
Other carrier frequencies may also be implicit in the GPS signals. For example, the m(t) wave is itself the product of several waves whose time intervals between polarity reversals are fixed integer multiples of tm. Thus, additional carriers whose frequencies are corresponding submultiples of 1/(2 tm) are implicitly present. One of the waveforms or factors multiplied together to produce m(t), known as the "C/A" code waveform or C/A code sequence, is a satellite-specific, pseudorandom, binary sequence of 1023 chips repeated periodically with a period of 1 millisecond, or a frequency of 1 kiloHertz.
Another factor in m(t) is a stream of binary "navigation" data having a 20-millisecond chip width, thus a 25 Hertz carrier frequency. These data include the current time indicated by the satellite's clock, a description of the satellite's current position in orbit, and a description of corrections to be applied to the time indicated by the satellite's clock. These data are broadcast by the satellites for use in the process of determining the position of a receiver from measurements of the received signals. Similar or identical data may be included in the n(t) wave which may modulate both the L1 and the L2 band center frequency carriers.
1.2.2.2. Carrier Reconstruction
Various techniques are known for reconstructing carrier waves from the spread spectrum signal received from a GPS satellite. In the conventional technique, the received signal is multiplied by a replica, generated locally, of the satellite-specific C/A code waveform present in m(t), or of the "P" or "Y" code which is present in n(t). In other techniques, no code sequence is generated in the receiver. Such codeless techniques may be utilized when the relevant code is unknown or to avoid code dependence.
An aspect of some carrier reconstruction techniques, including the codeless technique used in the preferred embodiment of the present invention described hereinbelow, is that the second harmonic rather than the fundamental frequency of an implicit carrier is reconstructed. In the preferred embodiment, second-harmonic frequencies of 616 f.sub.0 and 2 f.sub.0 are reconstructed from the GPS signals received in the L1 band, and frequencies of 480 f.sub.0 and 2 f.sub.0 are reconstructed from the signals received in the L2 band.
1.3. Deriving Position Information from GPS Signals
Various methods are known for deriving position information from a signal received from a GPS satellite. In some methods, the time delay of the pseudorandom code modulation of the signal is measured. In others, the phase of a periodic carrier wave implicit in the signal is measured. Time delay and carrier phase measurements may be combined. In any case, information relating both to the position of the receiver and to the position of the orbiting satellite is obtained. The present invention is primarily concerned with the determination of orbital position information.
1.3.1. Using Carrier Phase
The position information obtainable by measuring the phase of a GPS carrier wave, especially one of the relatively short wavelength, L1 or L2 band center frequency carriers, is potentially much more accurate than the information obtainable by measuring the modulation delay. However, the potential accuracy of carrier phase information can be difficult to achieve because carrier phase measurements are ambiguous. Their full potential cannot be realized unless the ambiguity problem can be resolved.
Because resolving phase ambiguity is an important aspect of the present invention, this problem and known methods of attacking or avoiding it are reviewed hereinbelow. The ambiguity problem is a fundamental one affecting all types of phase measurements, but its nature and the difficulty of solving it depend strongly on the techniques used to collect and process the measurements. The nature of the ambiguity problem, whether it can be solved, and if so how, depend particularly on how well the positions of the satellite and the receiving station are known. Uncertainty in knowledge of a satellite orbital position causes more serious difficulty in solving the ambiguity problem than uncertainty in a fixed receiver position.
Which position is unknown is critical because, for example, a fixed receiver position may be specified for the entire time span of an extensive set of observations, by the values of just three coordinates (for example, latitude, longitude, and height). On the other hand, a minimum of six parameters must be specified to define the orbit of a satellite, even for a relatively short time span.
Techniques are known for solving the carrier phase ambiguity problem in determining unknown receiver coordinates, but only when the relevant satellite orbital parameters are relatively well known. The most efficient techniques known for receiver position determination rely on a method of phase data processing known as "double differencing". In double difference phase processing, as described below, the problem of resolving carrier phase ambiguity appears as a problem of determining integer numbers called ambiguity parameters.
The present invention addresses the problem of phase ambiguity in the context of determining unknown orbital parameters. This problem, as mentioned, is much more difficult than the ambiguity problem in determining unknown receiver position coordinates.
To determine unknown orbital parameters, it is known to use double difference phase processing. When this is done, however, the orbital uncertainty interferes with determination of the integer values of the ambiguity parameters. Because the ambiguity parameters cannot be determined, in other words because the ambiguity of carrier phase is not resolved, the accuracy of the orbit determination is degraded.
The difficulty of resolving phase ambiguity in the orbit determination process is such that the usually recommended procedures do not include any attempt to resolve phase ambiguity
The present invention enables more accurate orbit determination by improving the ability to resolve phase ambiguity in the process. As an aid to understanding the invention, known methods of resolving ambiguity, usable in the determination of an unknown receiver position when orbits are already accurately known, are reviewed hereinbelow. Why known methods of resolving ambiguity fail when the orbits are unknown is also discussed.
1.3.1.1. Double Differencing
As mentioned, it is known to determine the position of a receiver by measuring the phase of a carrier wave implicit in signals received from a GPS satellite. The most accurate methods involve comparing the phases of the carrier waves of signals received simultaneously from different satellites. Carrier waves (or their second harmonics) are reconstructed from the received signals, and the phases of these carriers are measured with respect to a local reference oscillator in the receiver. The carrier phase measurement data are processed to determine position coordinates of the receiver.
Known methods of processing address the fact that carrier phase measurements are corrupted by additive biases. The biases stem from three sources: (1) The measured phase includes the phase of the transmitting oscillator in the satellite. This phase is not only random; it varies randomly with time. (2) The phase of the receiver's local oscillator has been subtracted from the measured phase. This phase also varies randomly with time. (3) In addition, the measured phase is biased by an unknown integer number of cycles because a carrier wave is a periodic wave. This integer cycle bias represents the inherent ambiguity of a carrier phase measurement.
Carrier phase measurements are ambiguous because a carrier wave is a periodic wave. One cycle of any periodic phenomenon is, by definition, indistinguishable from any other cycle. By observing a periodic wave such as a reconstructed GPS carrier signal continuously, one can determine its phase changes unambiguously. The total value of a phase change, including both the integer number of cycles and the additional fraction of a cycle, can be observed. However, without more information one cannot determine the initial value of the phase.
Because the initial value is unknown, a continuous series of phase measurements has an unknown, constant bias. As long as the bias is unknown, useful information can not be derived from the average, or mean, value of the series of measurements. Although useful information is contained in the variation about the mean, the mean value will only contain useful information if the bias can be determined.
The bias of a series of carrier phase measurements stemming from the phase of any given satellite's oscillator may be cancelled by subtracting measurements of that satellite's signal made simultaneously at two different receiving stations. The resulting between-stations difference observable is still useful for determining the position of one receiver if the position of the other is known.
The bias of a series of carrier phase measurements stemming from the phase of any given receiver's oscillator may be cancelled by subtracting simultaneous measurements by that receiver of two different satellites. The resulting between-satellites difference observable is still useful for determining the position of the receiver.
Biases related to both kinds of oscillators are canceled if both types of differencing are employed: between stations and between satellites. This is known as double-differencing, or doubly differenced phase processing.
The double differencing method requires a plurality of satellites to be observed simultaneously at each of a plurality of receiving stations. At each station, carriers are reconstructed from the received signals, and the carrier phases are measured with respect to the local reference oscillator, for all the satellites at the same time. Then differences are taken between phases measured for different satellites at the same time, in order to cancel the common errors associated with the local oscillator phase.
Carrier phase measurements from three or more receivers at a time may be combined in a double-differencing mode. If at a specific epoch, n receivers observed m satellites, then (n-1)(m-1) linearly independent double differences can be formed. An efficient algorithm for combining carrier phase data from more than two receivers is described in the article by Yehuda Bock, Sergei A. Gourevitch, Charles C. Counselman III, Robert W. King, and Richard I. Abbot, entitled "Interferometric Analysis of GPS Phase Observations", appearing in the journal manuscripta geodaetica, volume 11, pages 282-288, published in 1986. As disclosed hereinbelow, the present invention involves the combination, in a doubly differenced mode, of measurements made by three or more receivers.
1.3.1.2. Ambiguity Resolution
An important consequence of the cancellation of transmitter and receiver oscillator phase contributions in doubly differenced phase measurements is that the constant bias of a continuous measurement series (due to ignorance of the initial value) is an integer number of cycles of phase. Sometimes the value of this integer can be determined, so that distance- or other position-related information can be derived from the average value of a series of measurements. The process of determining the integer value of the bias of a series of phase measurements is called "resolving the ambiguity" of the series.
Because doubly differenced phase ambiguity resolution is an essential part of the present invention, the concept will be reviewed further as an aid to understanding the invention. The following review uses the notation and some of the language of an article by G. Beutler, W. Gurtner, M. Rothacher, T. Schildknecht, and I. Bauersima, entitled "Using the Global Positioning System (GPS) for High Precision Geodetic Surveys: Highlights and Problem Areas", appearing in the IEEE PLANS '86 Position Location and Navigation Symposium Record, pages 243-250, published in 1986 by the Institute of Electrical and Electronics Engineers, New York. For clarity, many details are omitted here.
Let L represent the wavelength of a reconstructed carrier wave, that is, the speed of light c divided by the reconstructed carrier frequency. In the case of a receiver which reconstructs the second harmonic of an implicit carrier frequency, the wavelength is computed from twice the implicit carrier frequency.
Let r.sup.i.sub.k represent the distance or "range" between receiver k at the reception and measurement time, t, and satellite i at the time of transmission, (t-r.sup.i.sub.k /c).
Let f.sub.k represent the phase of the k.sup.th receiver's local reference oscillator, and let f.sup.i represent the phase of the i.sup.th satellite's transmitting oscillator.
Then the so-called "one way" phase observable f.sup.i.sub.k, for the signal received from the i.sup.th satellite at the k.sup.th receiver, is given theoretically by the equation EQU f.sup.i.sub.k =f.sup.i -f.sup.i -f.sub.k -(1/L)r.sup.i.sub.k +N.sup.i.sub.k, tm (Eq. 1)
where all phases are expressed in cycles and N.sup.i.sub.k is an integer expressing the intrinsic ambiguity of this phase observable.
Four one-way phases measured at the same epoch t, at a pair of receiving stations k and q and for pair of satellites i and j, are differenced to form a doubly differenced observable: EQU DDf.sup.i.sub.k.sup.j.sub.q =(f.sup.i.sub.k -f.sup.i.sub.q)-(f.sup.j.sub.k -f.sup.j.sub.q). (Eq. 2)
Again, subscripts denote receivers and superscripts denote satellites. The double differencing cancels the transmitter and the receiver oscillator phases. The effects of the differences between the satellite-to-receiver distances, and a bias which is an integer number of cycles, remain: EQU DDf.sup.i.sub.k.sup.j.sub.q =-(1/L)DDr.sup.i.sub.k.sup.j.sub.q +N.sup.i.sub.k.sup.j.sub.q. (Eq. 3)
Here, DDr.sup.i.sub.k.sup.j.sub.q is the doubly differenced range, and N.sup.i.sub.k.sup.j.sub.q is the integer bias, sometimes called the "ambiguity parameter".
Determining uniquely the true integer value of the unknown bias of a continuous series of doubly-differenced phase observations is called "resolving the ambiguity" of the series. If the ambiguity parameter of a series can be determined, it may be subtracted from each observation in the series, or otherwise accounted for. Then useful information may be derived from the average value of the series of measurements. Thus, the value of an observation series is enhanced by determination of the bias.
In general, a series of observed values of doubly-differenced phase is composed of a mean, or average, value, plus a variation about the mean. Both the mean value and the variation about the mean contain potentially useful information about the positions of the satellites and the receivers. The mean value of the phase is related to the mean of the doubly-differenced satellite-receiver distance, and the variation of the phase is related to the variation of this distance.
If the mean value includes an additive bias which is unknown, then one does not know the value of the position-related part, so it is difficult to derive meaningful position information from the mean value. However, once the additive bias is known, the position-related part of the mean value of the observed phase is known and can contribute to determining the positions of the receivers.
If the positions of the satellites were unknown and the additive bias could be determined, the mean value of the observed phase could contribute to determining the positions of the satellites. Determining the additive bias and applying the mean value information to determine the positions of satellites is an aspect of the present invention.
One method of determining the integer bias of a series of doubly-differenced phase observations is simply to utilize sufficiently accurate information from an external source to calculate the value of the phase observable with an uncertainty of less than one-half cycle. A simple example of using information from an external source would be the use of independently derived information about the positions of the satellites and the stations to calculate the doubly differenced range, DDr.sup.i.sub.k.sup.j.sub.q, in Eq. 3. Substituting the actually observed value of the doubly differenced phase for the theoretical value, DDf.sup.i.sub.k.sup.j.sub.q, in Eq. 3 yields an equation which may be solved for the ambiguity parameter, N.sup.i.sub.k.sup.j.sub.q.
Another example of using independently derived information to determine the ambiguity parameter is the use of a "parallel" series of doubly-differenced observations, from the same pair of stations and for the same pair of satellites, and at one or more of the same measurement epochs, of the satellite-to-station path length as inferred from the time delay of the code modulation of a satellite signal. This method was proposed in a paper published in 1979 by C. C. Counselman III, I. I. Shapiro, R. L. Greenspan, and D. B. Cox, Jr., entitled "Backpack VLBI Terminal with Subcentimeter Capability", appearing in National Aeronautics and Space Administration Conference Publication 2115, "Radio Interferometry Techniques for Geodesy", on pages 409-414. A detailed development of this method was given in a paper by Ron Hatch, entitled "The Synergism of GPS Code and Carrier Measurements", appearing in the Proceedings of the Third International Geodetic Symposium on Satellite Doppler Positioning, volume 2, pages 1213-1231, published in 1982 by the Physical Science Laboratory of the New Mexico State University.
This method relies on the ability to determine the doubly-differenced range from observations of the modulation delay with sufficiently small uncertainty that the bias of the doubly-differenced center frequency carrier phase for the same station pair and satellite pair is computable with less than one-half cycle of error. An important aspect of this method is that it does not require determination or external knowledge of the geometry. The satellite-to-receiver distance, whatever its value, delays the signal modulation and the center-frequency carrier by the same amount. Therefore the ability to resolve ambiguities by this method is essentially independent of uncertainty in available knowledge of the station positions and the satellite orbits.
Unfortunately, it has proven extremely difficult in practice to measure the modulation delay of the signal with sufficient accuracy to ensure correct resolution of the L band center-frequency carrier phase ambiguities, and the utility of this method has so far been rather limited.
Related methods of resolving ambiguities in phase observations of GPS satellites are known in which phases are observed for a plurality of reconstructed carriers including one or more subcarriers. The phase of a subcarrier is indicative of modulation delay.
Methods of resolving ambiguities in which carrier phase observations are made at up to about ten different frequencies, including some closely spaced frequencies, some widely spaced frequencies, and some progressively spaced intermediate frequencies, are also known, as proposed for example by C. C. Counselman III and I. I. Shapiro in the paper entitled "Miniature Interferometer Terminals for Earth Surveying" published in the Proceedings of the Second International Symposium on Satellite Doppler Positioninq, Vol. II, pp. 1237-1286, January 1979, available from the University of Texas at Austin. This method is akin to the method of bandwidth synthesis employed for the unambiguous measurement of delay in very long baseline radio interferometry, as described in a publication by A. E. E. Rogers, entitled "Very Long Baseline Interferometry with Large Effective Bandwidth for Phase Delay Measurements", appearing in Radio Science, vol. 5, no. 10, pages 1239-1247, October 1970.
Simultaneous observation of different frequencies, and/or the combination of code delay and carrier phase measurements, is also known to be useful for the purpose of determining, and thereby eliminating, the frequency-dependent effects of ionospheric refraction on the satellite signals.
The known multiple-frequency and bandwidth synthesis methods are very much like the above mentioned GPS code-delay method; all are independent of, and do not involve knowledge or determination of, the satellite-station geometry. Unfortunately, the signals transmitted by the GPS satellites are not really suitable for use of the multiple-frequency and bandwidth synthesis methods. A serious problem is that the widths of the GPS L1 and L2 bands are too small in comparison with the frequency spacing between the bands. It is the relatively narrow GPS signal bandwidth which also severely limits the utility of the code-delay method. The reasons behind the limitation are related.
As discussed herein below, the determination of satellite orbital data in accordance with the present invention involves the use of at least three receiving stations preferably including some closely spaced stations, some widely spaced stations, and stations with a progression of intermediate spacings. The spacings in this case refer to geometrical distance. However, an analogy exists between the use of progressively spaced stations and the use of progressively spaced frequencies. Although it may not be feasible to equip the GPS (or any other) satellites to transmit a suitable progression of frequencies, it is indeed feasible to set up an array of tracking stations with a suitable progression of geometrical spacings. In a sense, therefore, the present invention may be said to compensate for the gaps in the GPS frequency spectrum which limit the use of known multi-frequency and related techniques.
Similarly, where a system provides a suitable spacing of frequency components, the dependence on varied base line lengths is reduced.
Of all known methods of resolving ambiguity in doubly-differenced phase observations, probably the most useful, and therefore most widely used in determining unknown station position coordinates when satellite orbital parameters are sufficiently accurately known a priori, is to estimate the ambiguity parameters and the station coordinates simultaneously by least-squares fitting to the doubly-differenced phase observations.
In this method the information which is contained in the variation about the mean of each series serves, in effect, to determine the unknown position-related quantities; from the determinations of these quantities the satellite-to-station path lengths are computed; the computed Path lengths are converted from distance to phase units by dividing by the wavelength, and are doubly differenced; the mean of the doubly differenced phase thus computed is subtracted from the actually observed mean; and the resulting difference is an estimate of the bias. Ideally this estimate is near an integer value and has sufficiently small uncertainty that the correct integer value of the bias can be identified with confidence.
In an extension of this method, every integer value in a finite interval surrounding the estimate of each ambiguity parameter (one for each continuous series of observations) is tested by repeating the least-squares adjustment, or "fit", of all the non-ambiguity parameters to the observations for each trial set of integer values of the ambiguity parameters. For each trial, the sum of the squares of the post-fit differences, or "residuals", between the observed and the corresponding computed values of doubly differenced phase is computed. This sum, which the least-squares fitting process attempts to minimize, indicates the badness of the fit. The particular set of integer values of ambiguity parameters found to have the smallest sum of squares of post-fit residuals is identified. Confidence in the correctness of this identification is indicated by the contrast between the related sum of squares, and the next-smallest sum or sums.
Ambiguity resolution by methods such as these is known to be useful in the processing of carrier phase data when the errors in the theoretically computed values of the phase observables are small in comparison with one cycle of phase. Obviously, if the magnitudes of these errors can approach or exceed one-half cycle, they can prevent the correct determination of the ambiguity parameters. It is known that such errors increase with increasing distance between a pair of receivers. The magnitudes of the phase errors are known to increase with increasing distance between the receiving stations for several reasons.
1.3.1.3. Effect of Orbital Uncertainty
One of the most important reasons is that an error in the assumed knowledge of a satellite's orbit causes an error in the theoretically computed value of a between-stations satellite range difference, such as Dr.sup.i.sub.kg for satellite i and stations k and q, which is proportional to the distance between stations k and q. The magnitude of the error is about equal to the inter-station distance multiplied by the orbital error measured in radians of arc as subtended at the midpoint of the baseline (and also as projected in a direction parallel to the baseline in question).
Thus, for example, if the orbital error as seen from a baseline on the ground and in the direction of the baseline is 2.times.10-7 radian, then the error in the computed value of Dr.sup.i.sub.kg will be 1 centimeter for a 50-kilometer distance, and 10 centimeters for a 500-kilometer distance. For observations of the L1 band center frequency carrier, which has a wavelength of about 19 cm, a 2.times.10-7 radian orbital error would probably not cause trouble in the ambiguity resolution process for a 50-km baseline. However, it might for a 500-kilometer baseline.
In general, it is known to use ambiguity resolution when the orbits of the satellites are known a priori with sufficient accuracy, and the distance between receivers is sufficiently short, that the phase error related to the orbital error is small in comparison with one-half cycle and therefore does not interfere with correct integer-cycle bias determination.
It is known to determine the orbits of GPS satellites by Processing doubly-differenced phase observations. But in this processing, as far as is known, doubly-differenced phase ambiguity resolution has not been practiced. The practice of doubly-differenced phase ambiguity resolution has been limited to the determination of unknown receiver positions when the orbits of the satellites have been known a priori with sufficient accuracy. Heretofore, whenever satellite orbits have been substantially unknown a priori, and doubly-differenced phase observations have been processed to determine the orbits, the unknown phase biases or ambiguity parameters have been estimated as real-number (i.e., numerically continuous, as opposed to integer or discrete-valued) unknowns along with the unknown orbital parameters.
Because the sensitivity of the between-stations differenced phase observable to orbital error increases with increasing distance between stations, it is known to use observations from receiving stations separated by the greatest possible distances in order to obtain the most accurate orbit determination. It is customary to use observations from stations separated by thousands of kilometers.
1.3.1.4. Avoiding Ambiguity Resolution
At least two methods of handling ambiguity parameters as continuous variables, rather than integers, are known. In both methods the variables representing the ambiguity, or continuous unknown bias, parameters, are real numbers like the variables representing the satellite orbits, etc. One method is to solve for the unknown ambiguity-related variables explicitly. That is, they are determined by solving a large set of simultaneous equations explicitly including all of the unknown variables. This solution yields estimates of the biases as well as estimates of the other unknowns. Performing such a simultaneous solution was the first step in one of the ambiguity resolution methods described above.
Another method avoids the whole matter of ambiguity parameters. In this method, known as the "implicit bias" method, the biases are eliminated, or solved for only "implicitly", by redefining the observable quantities so that they have no biases. Each series of doubly-differenced phase observations for a given station pair and satellite pair is replaced by itself minus the arithmetic mean, or average, value of the original series. If DDf(t.sub.i) represents the doubly-differenced phase observation at the i.sup.th epoch t.sub.i, the new, unbiased observation DDf'(t.sub.i) is given by EQU DDf'(t.sub.i) =DDf(t.sub.i)-Average of [DDf(t.sub.i)]. (Eq. 4)
This bias-cancelling operation is performed separately for each doubly-differenced observation series, that is, for each station/satellite pair. Now, ambiguity parameters do not appear at all in the set of equations which is solved to determine the orbital parameters, etc.
In this method, all position related information contained in the mean value of the original series of observations is thrown away when the mean is subtracted. Of course, the information is also wasted in the "explicit" bias determination method if the biases are treated as real numbers and never fixed at their integer values, i.e. if the ambiguities are not resolved. The advantage of the "implicit" method, if the ambiguities are not going to be resolved anyway, is a simplification of the computations, due to the reduction of the number of unknowns to be solved for.
Although there are great differences between the explicit and the implicit methods with respect to practical matters such as computer size, speed, and precision requirements, there is no theoretical difference between these methods regarding the accuracies of the non-bias parameter determinations, provided of course that ambiguity resolution is not considered. Because ambiguity resolution is generally not considered in GPS orbit determination, Beutler and others have recommended the implicit-bias method of processing doubly differenced phase measurements for orbit determination.
1.3.1.5. Orbit Determination
The use of doubly-differenced phase observations for GPS satellite orbit determination is disclosed in an article by R. I. Abbot, Y. Bock, C. C. Counselman III, R. W. King, S. A. Gourevitch, and B. J. Rosen, entitled "Interferometric determination of GPS satellite orbits", appearing in the Proceedings of the First International Symposium on Precise Positioning with the Global Positioning System, vol. 1, pages 63-72, published in 1985 by the National Geodetic Information Center, National Oceanic and Atmospheric Administration, Rockville, Md., 20852, U.S.A.
The principles and the practice of GPS satellite orbit determination from doubly differenced carrier phase data are further disclosed in an article by G. Beutler, W. Gurtner, I. Bauersima, and R. Langley, entitled "Modeling and estimating the orbits of GPS satellites", appearing in pages 99-112 of the same Proceedings volume, and in an article by G. Beutler, D. A. Davidson, R. B. Langley, R. Santerre, P. Vanicek, and D. E. Wells, entitled "Some theoretical and practical aspects of geodetic positioning using carrier phase difference observations of GPS satellites", published in 1984 as Technical Report No. 109 of the Department of Surveying Engineering, of the University of New Brunswick, Canada.
The refinement of station position coordinates and a priori satellite orbital parameters by adjusting both simultaneously to fit doubly-differenced phase observations has also been disclosed, for example in the paper by Gerhard Beutler, Werner Gurtner, Markus Rothacher, Thomas Schildknecht, and Ivo Bauersima, entitled "Determination of GPS Orbits Using Double Difference Carrier Phase Observations from Regional Networks", appearing in the Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, volume 1, pages 319-335, published in 1986 by the Applied Research Laboratories of the University of Texas at Austin.
However, the utilization of ambiguity resolution in GPS satellite orbit determination is not known. When the orbits have been substantially uncertain, specifically when the combination of orbital uncertainty and inter-station distance yields phase bias uncertainty approaching or exceeding one-half cycle, then it is not known how to determine the bias parameters with uncertainty small enough to permit unique identification of their integer values. If the explicit solution method is used to estimate the biases simultaneously with the orbital parameters, one tends to find that the uncertainties of the bias estimates are not much smaller than one cycle.
Analysis reveals that the relatively large uncertainties in the estimates of the bias parameters when these parameters are estimated simultaneously with orbital parameters results from the fact that a change in the estimate of a bias parameter may be masked very effectively by certain kinds of changes in the estimates of the unknown orbital parameters. That is, the orbit may be adjusted in a certain way, and the bias parameters also changed, such that the net effects on the calculated values of the doubly-differenced phase observables are less than the measurement uncertainties. In other words, it is theoretically possible to shift the orbit such that the observable quantity changes by a nearly constant amount--which resembles the effect of a change in the bias.
Accordingly, it is said that the bias parameters are difficult to separate from the orbital parameters. It is also said that the bias parameters are correlated with the orbital parameters. The difficulty of separating biases from orbital parameters is greater if the time span of the observations is shorter. However, the difficulty is substantial even if a satellite is observed for the duration of its visible "pass", from horizon to horizon. The difficulty is such that ambiguity resolution has not been considered feasible in the context of orbit determination.
From the difficulty in separating the bias and the orbital parameters, it follows that if some way could be found to determine uniquely the integer values of the biases, then the orbital parameters could be determined more accurately.